HMMT 十一月 2023 · THM 赛 · 第 2 题

HMMT November 2023 — THM Round — Problem 2

Suppose rectangle F OLK and square LORE are on the plane such that RL = 12 and RK = 11 . Compute the product of all possible areas of triangle RKL .

解析

Suppose rectangle F OLK and square LORE are on the plane such that RL = 12 and RK = 11. Compute the product of all possible areas of triangle RKL . Proposed by: Rishabh Das Answer: 414 Solution: There are two possible configurations, as shown below. K F E R E R K F L O L O √ If RL = 12, the side length of the square is 6 2. Now √ 2 2 2 2 2 121 = RK = RE + EK = (6 2) + EK , √ so EK = 7. Then the possible values of LK are 6 2 ± 7. Note that the area of △ RLK is √ LK · RE = LK · 3 2 , 2 and so the product of all possible areas are √ √ √ √ √ √ √ 2 3 2 (6 2 + 7) · 3 2 (6 2 − 7) = (6 2 + 7)(6 2 − 7) · (3 2) = (72 − 49) · 18 = 414 .